# Exact solutions of the Wheeler-DeWitt equation with ordering term in a dark energy scenario

## TL;DR

This paper provides exact solutions to the Wheeler-DeWitt equation in dark energy cosmologies, revealing how different ordering parameters influence the universe's initial conditions and ultimate fate, including big rip scenarios.

## Contribution

It analytically solves the Wheeler-DeWitt equation with ordering term for dark energy models, exploring initial singularity probabilities and end states like big rip.

## Key findings

- For q=1, universe likely evolves from a big bang singularity.
- For q=0, initial singularity is unlikely, universe starts finite.
- In phantom scenarios, universe ends in a big rip in finite time.

## Abstract

We investigate the quantum evolution of the universe in the presence of two types of dark energies. First, we consider the phantom class ($\omega<-1$) which would be responsible for a super-accelerated cosmic expansion, and then we apply the procedure to an ordinary $\Lambda>0$ vacuum ($\omega=-1$). This is done by analytically solving the Wheeler-DeWitt equation with ordering term (WdW) in the cosmology of Friedmann-Robertson-Walker. In this paper, we find exact solutions in the scale factor $a$ and the ordering parameter $q$. For $q=1$ it is shown that the universe has a high probability of evolving from a big bang singularity. On the other hand, for $q = 0$ the solution indicates that an initial singularity is unlikely. Instead, the universe has maximal probability of starting with a finite well-defined size which we compute explicitly at primordial times. We also study the time evolution of the scale factor by means of the Hamilton-Jacobi equation and show that an ultimate big rip singularity emerges explicitly from our solutions. The phantom scenario thus predicts a dramatic end in which the universe would reach an infinite scale factor in a finite cosmological time as pointed by Caldwell et al. in a classical setup. Finally, we solve the WdW equation with ordinary constant dark energy and show that in this case the universe does not rip apart in a finite era.